But in 2009 his wife died, and the business they owned together in Los Angeles fell apart. So Mr. Belgrade converted the structure and rented it out to help provide for himself and his young son.

All was well until about two years ago, he says, when city inspectors determined that the unit violated city ordinance because it was too close to the property line. Belgrade would have to apply for a variance, or an exemption from the city, which could take months and run into thousands of dollars – without any guarantee of approval.

“Regulation needs to come with safety. Is it a fire trap? Is someone going to die in there? Not, ‘Oh, it’s not 15 feet from the back [of the property],’ ” he says. “Who is this hurting?”

Secondary homes, or “granny flats,” have become a rallying cry in an ongoing battle over affordable housing. But Belgrade’s predicament is also indicative of a broader cultural clash, researchers and observers say. Longstanding notions of the ideal American home, and policies meant to preserve it, are colliding with a reality of soaring housing costs, stagnant incomes, and a growing desire to live close to walkable city centers. The result is a struggle that cuts to the heart of the American Dream as it was imagined by the generations following World War II.

“From about post-World War II to the ‘70s, cities became defined by single-family housing,” especially in the West, says Vinit Mukhija, a professor and vice chair of urban planning at the University of California, Los Angeles. “But so many factors seem to have changed [since then]. Land is much more expensive, transportation costs are much more expensive, family structure has changed.”

“Most important is the cost of housing is so high now,” he adds. “I think we may be reaching a point where [our perspective around housing] has to be rethought.”